Heat a large skillet over medium heat. Add coconut and toast it, stirring often, until golden brown, 2 to 3 minutes. Remove and set aside.

To make dressing: Heat a large skillet over medium heat. Add sesame seeds to skillet and toast, stirring constantly, until golden, about 2 minutes. Add 1/4 cup peanut oil and ginger; cook, stirring often, until very fragrant, 2 to 3 minutes. Transfer to a large, heatproof bowl and whisk in lime juice, fish sauce, and sugar; set aside.

In a medium bowl, toss shallots with flour; shake off excess flour. Heat remaining 1/4 cup peanut oil in a large skillet over medium-high heat, about 30 seconds. Add shallots and cook, stirring often, until deep golden brown, 5 to 7 minutes. Line a plate with paper towels. Using a slotted spoon, transfer shallots to plate and season with salt to taste; set aside to cool.